Kirchhoff's Voltage Law (KVL)

Algebraic sum of potential differences around any closed loop is zero. Statement of energy conservation. Sign convention: potential rises across EMF sources, drops across resistors in the direction of current.

Statement

The algebraic sum of all potential differences (EMFs and resistive drops) around any closed loop is zero:

$$\boxed{\sum_{\text{loop}} V = 0}$$

Basis: energy conservation

The electrostatic field is conservative — the work done per unit charge around any closed path is zero:

$$\oint \vec{E}\cdot d\vec{l} = 0$$

This is equivalent to KVL. A charge traversing a complete loop returns to the same potential — energy gained from EMF sources equals energy dissipated in resistors.

Sign convention

Traversing a loop in a chosen direction:

• Resistor in the direction of assumed current: $-IR$ (potential drop)
• Resistor against the direction of assumed current: $+IR$ (potential rise)
• EMF source from $-$ to $+$ terminal: $+\mathcal{E}$ (potential rise)
• EMF source from $+$ to $-$ terminal: $-\mathcal{E}$ (potential drop)

Counting equations

For a circuit with $b$ branches and $j$ junctions: there are $b - (j-1)$ independent KVL equations. Total equations from KCL + KVL = $b$, matching the number of unknowns (branch currents).